Multi-Variable Calculus Assignment

due Friday, February 2, 2001

Problems

1. Let A, B, and C be the points in R^{3} given by

A = (2, −3, 5) , B = (−1, 0, −1) , C = (3, 1, 4) ,

and let Δ be the triangle with these points as vertices. Find:

(a) The parametric representation of the line through B and C that assigns parameter value 0 to the point B and parameter value 1 to the point C.

(b) The midpoint M of the line segment BC.

(c) The point on BC that is one-third of the way from B to C.

(d) The parametric representation of the line through A and M that assigns parameter value 0 to A and parameter value 1 to M.

(e) Recall from plane geometry where the median of Δ is located on AM, and then

determine that point.

2. Two lines L and M in space are parameterized, respectively, by functions F and G that are given by the formulas

L: F (t) = (3t, −t, t + 2) M: G(t) = (2t − 1, −t + 1, t + 1) .

(a) Determine the points a on L and b on M obtained from parameter value 0 using F and G, respectively.

(b) Determine the vector v along L and w along M that runs, in each case, from the point where t = 0 to the point where t = 1.

(c) Find the angle between the vectors v and w.

(d) Find the “cross product” u = v × w.

(e) Compute u · F (t) and u · G(t).

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